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Critical Inequalities and Their Role in Forming the Set Ψ1 in L-Function Theory by@eigenvalue

Critical Inequalities and Their Role in Forming the Set Ψ1 in L-Function Theory

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Eigen Value Equation Population

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June 2nd, 2024
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The section defines the set Ψ1 and establishes key inequalities (3.4, 3.5, 3.6) using lemmas and Cauchy's inequality, crucial for understanding Dirichlet L-functions.
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STORY’S CREDIBILITY

Academic Research Paper

Academic Research Paper

Part of HackerNoon's growing list of open-source research papers, promoting free access to academic material.

Author:

(1) Yitang Zhang.

  1. Abstract & Introduction
  2. Notation and outline of the proof
  3. The set Ψ1
  4. Zeros of L(s, ψ)L(s, χψ) in Ω
  5. Some analytic lemmas
  6. Approximate formula for L(s, ψ)
  7. Mean value formula I
  8. Evaluation of Ξ11
  9. Evaluation of Ξ12
  10. Proof of Proposition 2.4
  11. Proof of Proposition 2.6
  12. Evaluation of Ξ15
  13. Approximation to Ξ14
  14. Mean value formula II
  15. Evaluation of Φ1
  16. Evaluation of Φ2
  17. Evaluation of Φ3
  18. Proof of Proposition 2.5

Appendix A. Some Euler products

Appendix B. Some arithmetic sums

References

3. The set Ψ1

Let ν(n) and υ(n) be given by


image


respectively. It is easy to see that


image


Lemma 3.1. Assume (A) holds. Then


image


Proof. Let


image


which has the Euler product representation


image


For σ ≥ σ0 > 0, by checking the cases χ(p) = ±1 and χ(p) = 0 respectively, it can be seen that


image


and


image


the implied constant depending on σ0. Thus φ(s) is analytic for σ > 1/2 and it satisfies


image


for σ ≥ σ1 > 1/2, the implied constant depending on σ1. The left side of (3.2) is


image


image


Lemma 3.2. Assume (A) holds. Then we have


image


Proof. As the situation is analogous to Lemma 3.1 we give a sketch only. It can be verified that the function


image


is analytic for σ > 1/2 and it satisfies


image


for σ ≥ σ1 > 1/2, the implies constant depending on σ1. Also, one can verify that


image


This completes the proof.


Lemma 3.3. For any s and any complex numbers c(n) we have


image


and


image


Proof. The first assertion follows by the orthogonality relation; the second assertion follows by the large sieve inequality.


Let


image


By (3.1) we may write


image


By Cauchy’s inequality and the first assertion of Lemma 3.3 we obtain


image


Thus we conclude


Lemma 3.4. The inequality


image


Write


image


Assume that (A) holds. By Cauchy’s inequality, the second assertion of Lemma 3.2 and Lemma 3.1,


image


Thus we conclude


Lemma 3.5 Assume that (A) holds. The inequality


image


Let


image


Assume that (A) holds. By Cauchy’s inequality, the first assertion of Lemma 3.2 and Lemma 3.1,


image


Thus we conclude


Lemma 3.6. Assume that (A) holds. The inequality


image


We are now in a position to give the definition of Ψ1: Let Ψ1 be the subset of Ψ such that ψ ∈ Ψ1 if and only if the inequalities (3.4), (3.5) and (3.6) simultaneously hold.


Proposition 2.1 follows from Lemma 3.4, 3.5 and 3.6 immediately.


This paper is available on arxiv under CC 4.0 license.


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